How can I automatically generate vertex groups for an armature?
If you first select the mesh and then the armature and use Ctrl + P > Armature Deform > With Automatic Weights, this will generate a vertex group for the part of the mesh that corresponds to the bone in the armature. (this is dependent on how many bones are in the armature and or how much geometry the mesh has)

You can then use either Edit or Weight Paint mode to tweak what parts of your mesh are affected by each group, to quickly select a group and see the selection, you can go to the Vertex Groups panel in the Object Data tab.



Additionally, you can also parent the mesh to the armature as empty groups..



and then select the bone while in Pose Mode, open the Specials menu with W or from the weight paint view header and choose Assign Automatic from Bones, this can give you a bit more control over how your groups are created as you can see it on the fly and edit it further.